# Mark Haskins

- Professor of Mathematics

**External address:**187 Physics Building, 120 Science Drive, West Campus, Durham, NC 27708-0320

**Internal office address:**120 Science Drive, 117 Physics Building, Campus Box 90320, Durham, NC 27708-0320

### Research Areas and Keywords

##### Analysis

Geometric measure theory, especially regularity theory and singularities of calibrated currents. Nonlinear systems of elliptic PDEs. Analysis on noncompact complete spaces and on incomplete or singular spaces.

##### Geometry: Differential & Algebraic

Special and exceptional holonomy spaces, especially Calabi-Yau 3-folds, $G_2$ holonomy and $Spin_7$ metrics. Calibrated submanifolds and currents in special holonomy spaces: special Lagrangian, associative and coassociative and Cayley submanifolds. Singular calibrated currents especially calibrated cones. Einstein and Ricci-flat metrics.

##### Mathematical Physics

M-theory and its connection to $G_2$ holonomy spaces.

##### PDE & Dynamical Systems

Nonlinear systems of elliptic PDEs. Regularity theory of calibrated currents. Analysis on noncompact complete spaces and on incomplete or singular spaces.

My research concerns problems at the intersection between **Differential Geometry** and **Partial Differential Equations**, particularly special geometric structures that arise in the context of holonomy in Riemannian geometry. Currently I am particularly interested in special types of 7-dimensional spaces called **G _{2}-holonomy manifolds**, or G

_{2}-manifolds for short. These spaces also arise naturally in modern theoretical physics in the 11-dimensional theory known as M theory. To get from 11 dimensions down to 4 dimensions it is necessary to 'compactify' on a 7-dimensional space and to preserve the maximal degree of (super)symmetry this 7-dimensional space should have G

_{2}-holonomy. In fact realistic 4-dimensional physics appears to demand singular G

_{2}-holonomy spaces and trying to construct compact singular G

_{2}-holonomy spaces is one of my current research projects.

Manifolds with special holonomy also come equipped with special submanifolds, called **calibrated submanifolds**, and special connections on auxiliary vector bundles, called generalised instantons. I am particuarly interested in **associative** and **coassociative submanifolds** in G_{2}-holonomy spaces and **special Lagrangian submanifolds** in Calabi-Yau spaces. In the past I have also studied *singular* special Lagrangian n-folds.

I am currently the Deputy Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics. My colleague here at Duke, Robert Bryant, is the Collaboration Director.

### Selected Grants

Special Holonomy in Geometry, Analysis, and Physics awarded by Simons Foundation (Principal Investigator). 2016 to 2020

Foscolo, L., and M. Haskins. “New G2-holonomy cones and exotic nearly Kahler structures on S6 and S3 x S3.” *Annals of Mathematics*, vol. 185, no. 1, Jan. 2017, pp. 59–130. *Scopus*, doi:10.4007/annals.2017.185.1.2.
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Corti, A., et al. “G2-Manifolds and associative submanifolds via semi-fano 3-folds.” *Duke Mathematical Journal*, vol. 164, no. 10, Jan. 2015, pp. 1971–2092. *Scopus*, doi:10.1215/00127094-3120743.
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Mark, H., et al. “Asymptotically cylindrical Calabi-Yau manifolds.” *Journal of Differential Geometry*, vol. 101, no. 2, Jan. 2015, pp. 213–65.
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Corti, A., et al. “Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds.” *Geometry and Topology*, vol. 17, no. 4, July 2013, pp. 1955–2059. *Scopus*, doi:10.2140/gt.2013.17.1955.
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Haskins, M., and N. Kapouleas. “The geometry of SO(p) × SO(q)-invariant special Lagrangian cones.” *Communications in Analysis and Geometry*, vol. 21, no. 1, Apr. 2013, pp. 171–205.

Haskins, M., and N. Kapouleas. “Closed twisted products and SO(p) × SO(q)-invariant special Lagrangian cones.” *Communications in Analysis and Geometry*, vol. 20, no. 1, Jan. 2012, pp. 95–162. *Scopus*, doi:10.4310/CAG.2012.v20.n1.a4.
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Haskins, M., and N. Kapouleas. “Special Lagrangian cones with higher genus links.” *Inventiones Mathematicae*, vol. 167, no. 2, Feb. 2007, pp. 223–94. *Scopus*, doi:10.1007/s00222-006-0010-5.
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Haskins, M., and T. Pacini. “Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem.” *Geometry and Topology*, vol. 10, Oct. 2006, pp. 1453–521. *Scopus*, doi:10.2140/gt.2006.10.1453.
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Haskins, M. “Special Lagrangian cones.” *American Journal of Mathematics*, vol. 126, no. 4, Aug. 2004, pp. 845–71.

Haskins, M. “The geometric complexity of special Lagrangian T2-cones.” *Inventiones Mathematicae*, vol. 157, no. 1, July 2004, pp. 11–70. *Scopus*, doi:10.1007/s00222-003-0348-x.
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